Where the points actually are

The jump from AMC to AIME is real. AMC problems reward recognition; AIME problems reward technique. You need a working toolkit of named methods, and you need to execute long computations without error, because every answer is all-or-nothing.

Topic breakdown

01Algebra: Polynomial roots and symmetric functions, complex numbers and roots of unity, sequences, functional equations, inequalities.
02Combinatorics: Recursion, generating functions at an introductory level, inclusion-exclusion, bijections, probability with conditioning.
03Number theory: Modular arithmetic to a serious level, Chinese remainder theorem, orders and primitive roots, Diophantine equations, base representation.
04Geometry: Power of a point, similar triangle configurations, trigonometric identities, coordinate and complex-number bashing, mass points.

A study plan that works backwards from the date

Immediately after the AMC You have roughly ten to twelve weeks. Take one past AIME cold to see where you stand. A score of 3 to 5 on your first attempt is normal and not a bad sign.
Weeks 1 to 6 Work through AoPS Intermediate Algebra and Intermediate Counting and Probability, or the Volume 2 book, one chapter every three or four days. Do the AIME-tagged problems in the Contest Collections after each chapter.
Weeks 7 to 10 Two full past AIMEs per week under timed conditions. Aim to finish problems 1 to 8 in the first 75 minutes so you have real time for 9 to 15.
Final two weeks Stop learning new material. Drill accuracy: redo problems you got right but slowly, and rebuild your arithmetic discipline.

Tips that come from the format itself

There is no partial credit and no guessing penalty. Fill in something for every question, even an educated estimate.
Problems 1 to 5 should be almost automatic. If they are not, more AMC practice will raise your AIME score faster than harder material will.
Write every step. Three-hour papers are lost to arithmetic, not to ideas.
Learn to abandon. Two problems solved cleanly beat four problems attempted and botched.

Practice resources

AoPS AIME problem archive Past problems · free
Every AIME since 1983 with solutions.
AoPS Intermediate series Book · paid
Intermediate Algebra and Intermediate Counting and Probability are the standard AIME preparation texts.
Evan Chen's olympiad handouts Notes · free
Free notes from an IMO gold medallist. Aimed slightly above AIME but excellent for geometry technique.
AoPS Contest Collections Practice · free
Searchable and filterable by contest and difficulty.

Past papers and official materials

AIME problems and solutions, all years
Free complete archive with answer keys.

Videos worth your time

AoPS contest walkthroughsArt of Problem Solving
AIME problem sets solved in full.

Skills this builds

Read what other students said about preparing